Find the length of the projection of the line segment joining the points P(3,-1,2) and Q(2,4,-1) on the line with direction rations `lt - 1, 2,-2 gt `.

Answer

To find the length of the projection of the line segment joining the points P(3, -1, 2) and Q(2, 4, -1) on the line with direction ratios (1, 2, -2), we can follow these steps:

### Step 1: Find the vector PQ
The vector \( \vec{PQ} \) can be calculated by subtracting the coordinates of point P from the coordinates of point Q.

\[ \vec{PQ} = \vec{Q} - \vec{P} = (2 - 3) \hat{i} + (4 - (-1)) \hat{j} + (-1 - 2) \hat{k} \]
\[ \vec{PQ} = -1 \hat{i} + 5 \hat{j} - 3 \hat{k} \]

### Step 2: Find the direction vector of the line
The direction vector \( \vec{d} \) of the line with direction ratios (1, 2, -2) is given as:
\[ \vec{d} = \hat{i} + 2\hat{j} - 2\hat{k} \]

### Step 3: Find the magnitude of vector PQ
The magnitude of \( \vec{PQ} \) is calculated using the formula:
\[ |\vec{PQ}| = \sqrt{(-1)^2 + (5)^2 + (-3)^2} \]
\[ |\vec{PQ}| = \sqrt{1 + 25 + 9} = \sqrt{35} \]

### Step 4: Find the magnitude of the direction vector
The magnitude of the direction vector \( \vec{d} \) is:
\[ |\vec{d}| = \sqrt{(1)^2 + (2)^2 + (-2)^2} \]
\[ |\vec{d}| = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Step 5: Find the dot product of vectors PQ and d
The dot product \( \vec{PQ} \cdot \vec{d} \) is calculated as follows:
\[ \vec{PQ} \cdot \vec{d} = (-1)(1) + (5)(2) + (-3)(-2) \]
\[ \vec{PQ} \cdot \vec{d} = -1 + 10 + 6 = 15 \]

### Step 6: Find the length of the projection
The length of the projection of \( \vec{PQ} \) onto \( \vec{d} \) is given by the formula:
\[ \text{Length of projection} = \frac{\vec{PQ} \cdot \vec{d}}{|\vec{d}|} \]
Substituting the values we found:
\[ \text{Length of projection} = \frac{15}{3} = 5 \]

### Final Answer
The length of the projection of the line segment joining points P and Q on the line with direction ratios (1, 2, -2) is **5**.

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